use of load depend ent ritz vectors in modal …

TO

A

Twrbidc

vic K

Eaha

dw Iapiie

b TtaebbC

The 14th

WoOctober 12-1

USE O

1 Ph.D 2 Disting

ABSTRACT

The performawidespread uresponse. Mobut it fails in is to present tdynamic forccalculated ressystems whervectors, in cinclusion of choosing ade

KEYWORDS

1. INTROD

Evaluation ofanalyzed usinhowever, it isa simple methstructures (mdistribution owith consider

In order to oattractive feaprocedures win the modal inertia force expansion aresystems. A sbuilding is pr

The accuracythe conditionapproximate evaluation ofbuildings, thebuildings to Chintanapakd

orld Confere17, 2008, Beij

OF LOAD

D. Student of guished Profes

T:

ance of a struse in perforodal Pushoversome cases othe use of Lo

ce; instead ofsponse of irrere higher mo

case of stiffehigher modequate number

S: Modal Pus

DUCTION

f structural reng either lineas very time cohod has been

mainly regulaof load, goodrable higher-m

overcome thistures, for inst

with invariant expansion odistribution

e then combistep-by-step sresented in Re

y of MPA muns under whicprocedure, M

f “regular” bue next step ia degree o

dee [2] have

ence on Earthijing, China

D DEPEND

Hossein K

f EQ. Eng., Islssor, Earthqu

ComplEmail: hk

ructural systermance basedr Analysis (M

of the irregulaoad Dependenf commonly uegular systemode effects car lower storie effects witr of required

shover Analys

esponse is a kar or nonlineaonsuming andn preferred in ar structures)d estimates womode contribu

s limitation, Mtance, it retaiforce distribuf the effectivfor each mo

ined by the Csummary of eferences [2]

ust be evaluatech it is appli

MPA has been uildings. Becas to determinf accuracy wshown that:

hquake Engi

DENT RITAN

Kayhani1 and

lamic Azad Uake Engineerex; Islamic A

kayhani@gma

m can be estd seismic desMPA) has beear structural snt Ritz vectorused eigen-m

ms when limitannot be ignoies, increase hout really cRitz vectors f

sis, Ritz vecto

key concept inar methods. N

d may not be practice. Alth, due to limould not be pution to respo

MPA proceduins the conceputions. In the ve earthquakeode. These “mCQC rule to o

the MPA proand [3].

ed for a wideicable for sei shown to beause vertical ine whether owhich is con“The MPA p

ineering

1

TZ VECTNALYSIS

d Mohsen G

University, Sciring and Struczad Universit

ail.com, mohs

timated usingsign due to ien suggested ystems (i.e. srs (LDR) whi

mode shape inted number ofored. The num

the accuracycomputing thfor vertically

ors, LDR, No

n performancNonlinear dynsuitable for ehough it may

mitations in ipromised for onse.

ure has been ptual simpliciMPA procedu

e forces is detmodal” demaobtain an estiocedure to e

e range of strusmic evaluati accurate enoirregularities

or not the MPnsidered suffprocedure is

TORS IN M

Ghafory-Ash

ience & Reseactural Departty, Tehran, Irasen@ashtiany

g a non-lineaits simplicityto increase th

stiffer lower sich takes into n the MPA in f modes is tomerical resuly of force rehem). There regular or irr

nlinear Static

ce based seismnamic analysiveryday engi provide goodits fundamenall types of s

proposed byity and compure, the seismtermined by nands due to imate of the stimate the s

uctural systemion of structu

ough in estimsignificantly PA can estimficient for prless accurate

MODAL P

htiany2

arch Complextment of Scienan y.com

ar static analyy in estimatinhe accuracy otories). The oaccount the order to imp

o be considerelts have indicesponse signare also som

regular structu

c Analysis, St

mic engineeriis is considerneering practd approximat

ntal concepts structures esp

y Chopra [3] putational easmic demand du

nonlinear stathe first fewtotal seismic seismic dema

ms and grounures. By studating seismicinfluence the

mate seismic ractical applie relative to t

PUSHOVE

x,Tehran, Irannce and Resea

ysis which hang inelastic sof Pushover objective of thspatial distrib

prove the acced, especiallycated that usinificantly (becme suggestionures considere

tiffness Irregu

ing. Structurered as ‘exact’ tice; the NSP tion for some

like the firpecially for st

which offerse of current pue to individu

atic analysis uw terms of th

demand for ands for a m

nd motions todying the biac demands fore seismic demdemands of ication? Chothe reference

ER

narch

as found structural analysis; his paper bution of curacy of y for stiff ing LDR cause of ns about ed.

ularity

es can be method, which is types of

rst mode tructures

s several pushover ual terms using the he modal

inelastic multistory

o identify as of this r seismic

mands on irregular

opra and e regular

TO f

m T

ft 2

Itlodlmia

Imf

ufani 3 Ti

A

The 14th

WoOctober 12-1

frame in estisoft-and-weakmodes it is su

The objectivestiffer lower frames determthe use of LD

2. PROPOSEIGENV

In 1982 Wilstook into accless computator LDR (Loadistribution oload vector,mode-accelerimpossible toanalogous to

− Static

− Pseud

− LDR

In each of themanner by firfirst vector costatic solutionusually consiforce or dispaccurate dispnoted that thein the continu

3. NUMER

To evaluate tin this researstories. Both AISC-89 (AS


imating the sk lower half;

usceptible to l

e of the propostories for the

mined by MPADR vectors in

SED METHOVECTOR

son et al. [8] count the effetional time th

ad Dependentof excitation){ }( )f s can

ration procedo generate vecpseudo static

c correction

do static vect

R vectors

e first two sumrst term or inorresponds ton. This is theidered, and implacement resplacement rese first pseudouation of the p

RICAL ANAL

the effectivenrch: 1) Regul

type of systSD method)


eismic dema; stiff, stronglack of accura

osed method ie estimation o

PA procedure wMPA procedu

OD: MPA U

introduced aects of spatialhan use of exat Ritz) vectors, the first Ritbe compute

dure). Since ctors which a

c vector used i

{ ( )}U t =∑or

{ ( )}r

iU t

=

= ∑

{ } [ ]sU K=

{ ( )}r

i

U t=

= ∑

mmation metn a static mano static solutioe major benemplementing sponse calcusponse do noto static vectorpaper, only th

LYSIS

ness of using ar moment reems were defor PGA=0.3

hquake Engi

nds of frame, or stiff-andacy for the ca

is to improve of the seismicwith the “exaure can impro

USING LDR O

a new dynaml distribution

act eigenvectos. Consider thtz vector whid as: 1{ }X =

this vector ware not excitein the “mode-

1

{ } ( )r

i ii

y tϕ=

+∑

1{ } ( )

r

i iy tϕ=

+∑1] { ( )}f s−

*

2

{ } ( )r

i iX y t=∑

thods the full nner by the seon and additioefit of using

LDR vectorlation (displat guarantee ar, { }sU , is idhe results for L

LDR or { sUesisting framesigned accor3g, soil type

ineering

2

es with strong-strong lower

ases in which

the accuracyc demands. Cact” nonlinearove the accura

OR PSUEDO

mic analysis mof the dynam

ors known as he input forcech is the stat

1[ ] { }K f− (whwill be useded by the assu-acceleration”

1[ ] ({ (K f s−+

1([ ] [ rK K−+ −

1

1( [ ]T

XX M X

+

loading vecto

econd term, wonal vectors rLDR vectorss ensures theacement respaccurate forcedentical to theLDR has been

}s vectors; twe systems andding to Irani

e III and Res

g or stiff-andr half”. As thhigher-mode

y of MPA for tComparison or response hisacy of MPA f

O STATIC V

method basedmic loading aWYD Ritz vee as { ( , )}F s ttic response ohich is knowd for extractiumed loading” based respo

)} { ( )})rs f s−

1] ){ ( )}r f s g−

*11/2

1

( ))

y tX

or has been awhich is not threpresent dynas in MPA appe consideratioponse converge responses de starting vecn presented.

wo types of sd 2) Irregularan National B

sponse reduct

d-strong first his method cocontribution

the vertically of the seismicstory analysisfor the case of

VECTOR IN

d on Rayleighand yielded mectors (Wilso

{ ( )}. ( )f s g t=

of the time inwn as pseuon of other

g. The obtaineonse spectrum

) ( )g t

( )g t

accounted for he case for LDamic contribuproach whereon of higher-mges faster tha

due to higher tor of LDR v

structural systr frame systeBuilding codtion factor R

story; soft, wonsiders the cannot be neg

irregular fram demands in

s response, shf irregular stru

STEAD OF

h-Ritz methodmuch better ron, Yuan and D

({ }( )f s is thndependent podo static vevectors it w

ed response fm approach [5]

either in the DR vectors; sution neglectee first few mmode participan force respmodes). It sh

vector approa

tems were coems with stiffde which is siR=6. The reg

weak, or first few glected.

mes with irregular

hows that uctures.

d, which results in Dickens) he spatial ortion of ector in

would be format is ].

dynamic since the ed by the

modes are pation in ponse so hould be ch. Thus

onsidered fer lower imilar to ular and

TO i

F0c

Caw Ftcrp

The 14th

WoOctober 12-1

irregular fram

For the input0.7g (to ensucontribution o

Considering tand pseudo swith nonlinea

Figure 3 showthe evaluatiocalculating drequired in Mproperties are

T1=0.98 sec T2=0.361 T3=0.211


mes and their

Figure 1 Regu

ground motiure the nonlinof higher mod

Fig

the effect of htatic vector foar time history

ws sample mn of two app

damping in eaMPA approache in the veloci

T1=1.49 sec T2=0.554 T3=0.323


respective pe

ular and Irregula

ion, a set of nnear responsedes has been u

gure 2 Spectral

higher modesfor such irreguy analysis wh

odal propertieproaches (LDach mode forh. It is importity sensitive r

T1=1.91sec T2=0.71 T3=0.42

hquake Engi

eriods for the

ar frame models

near-field groue) with impuused.

accelerations for

s on responseular systems hich is assume

es for the 8-sDR vectors an

r the analysistant to considregion or not.

ineering

3

first 3 modes

that have been u

und motionslsive characte

r selected groun

e for the case (also for regued to be exact

story irregularnd eigenvectos of SDOF (S

der rational da Different sys

T1=0.T2=0.2T3=0.

T1=2.35 secT2=0.888 T3=0.531

are shown in

used in the appli

recorded on ser which can

d motions and m

of stiffer lowular systems ft response) ha

r frame whichors). RayleighSingle Degreamping ratiosstems have be

62 sec 205 099

T1=0T2=0T3=0

c

n Figure 1.

ication of MPA w

soft soil (typen cause consi

mean spectrum

wer stories, thfor measuringave been exam

h have been oh damping ratee of freedoms since it is noeen coded acc

0.87 sec 0.311 0.183

T1=1T2=0T3=0

with LDR

e 3) scaled toderable respo

e use of LDRg the accuracymined.

obtained and tio has been

m) systems wot known whecording to the

1.23 sec 0.442 0.261

T1=T2=0T3=0

o PGA of onse and

R vectors y, in line

used for used for

which are ether the e number

1.6 sec 0.558 0.334

TO oeton

Fcb

f 3 Ie

The 14th

WoOctober 12-1

of stories, reexample 4sIrthat since speof each compnumber of LD

Figure

Figure 4(a) scan be seen thbetween 3rd estiffer part offirst few mod

3.1. Story Sh

In the case eigen-modes


egularity, typer-e(2) represeectral content puted vector DR vectors ca

Figure 3 S

4 Comparison o

hows the diffhere is no sig

eigenvector anf the structuredes.

hear and Mom

of shear respshould be co


e of vectors nt the resultsof the first vedepends on t

an significant

Sample Modal pr

of Eigenvectors a

fference betwegnificant partind 3rd LDR ve and as a res

ments

ponse of irreonsidered for

hquake Engi

(LDR or eigs for a 4 storyector in LDR the requestedly reduce the

roperties for 8 st

(a) and LDR vector

een First LDRcipation of loector (in a se

sult their cont

egular structur 16-story irr

ineering

4

genvector) any Irregular-frvector appronumber of Lcomputation

tory irregular fra

rs (a) First LDRv

R vector or Power stories int of three).It itribution to th

ure, to satisfregular frame

d number of ame, using 2

oach is spreadLDR vectors. nal effort need

ame for eigen-mo

(b) vector vs. 1st eig

Pseudo static n either case. is apparent thhe final respo

fy 90% of me but in case

f modes that eigen-modes

d among all thAppropriate

ded for MPA.

odes and LDR v

envector (b) 3rd

vector and fiFigure 4(b) s

hat LDR vectoonse can be ca

mass participof using LD

have been us). It should b

he basis vectorselection of

vectors

vector comparis

irst eigenvectshows the comors have activaptured by us

pation requireDR vectors, 3

used; For be noted rs, shape required

son

tor. As it mparison vated the sing only

ement, 8 3 vectors

TO we

otb

Fnev

The 14th

WoOctober 12-1

would be sueigenvector wshear of 16 sonly cause 10the higher mobut fewer vec

Figure 6 showno noticeableeigenvectors.vectors than superiority of


ufficient. Figuwith respect totory irregular0% error. Thiodes. In case ctors would b

Figure 5 Com

Table 1

Story

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

ws the similare benefit due As the struceigen-modes,f LDR vector


ure 5 and Tao nonlinear tir structure aftis trend can bof the regula

e needed to s

mparison of stor

MPA error perc

4 Story-IrreguEigen LD

- -- -- -- -- -- -- -- -- -- -- -- -

-5.14 7.6-1.76 0.9-25.55 -8.-45.27 -26

r results for tto the fact th

ctures get tall, still there iss is noticeabl

hquake Engi

able 1 show ime history anter using 4 ei

be seen for stiar structures uatisfy 90% pa

ry shear between

centage in Story

ular 8 Story-DR Eigen - - - - - - - - - - - - - - - - - 11.85 - 2.16 - 4.19 - 5.86 67 1.70 93 -17.96 .75 -31.24 6.34 -43.31

he story momhat LDR vecler, more accus no significane.

ineering

5

the advantanalysis (NLTHigen-modes wiffer part of stuse of LDR varticipation of

n NLRHA and M

Shear response

-Irregular 12LDR E

- - - - - -1- -7- -- 1

10.24 46.99 84.70 15.43 64.27 --8.91 -1-19.16 -3-31.02 -4

ments. Using ctors for theseurate results nt difference.

age of MPA HA) for vario

will have 40%tructures becavectors does nf mass.

MPA using eigenv

using LDR vect

2 Story-IrregularEigen LDR

- - - - - - - -

14.55 -22.137.88 -7.291.57 -1.831.74 1.134.95 5.358.01 8.880.00 10.416.76 7.106.52 0.0319.09 -5.5231.54 -13.8941.70 -23.69

LDR vectors e kinds of strcan be obtain. In the case

analysis witous types of f

% error while ause these panot cause sign

vectors and LDR

tors or eigenvect

r 16 Story-Irr Eigen

-4.48 2.47 1.57 2.76

3.97 5.79 6.84 9.12 11.08 9.72 1.63 -4.64

-14.55 -24.86

-35.42 -43.67

for low-rise ructures rapidned by using of irregular f

th LDR vectframes. The fiusing 3 LDR

arts are not exnificant impr

R vectors

tors

regular LDR

-14.16 1.58 2.38 0.45 3.06 8.14 10.38 11.71 11.56 10.01 6.55 7.12 9.49 7.56 -0.26 -9.59

regular structdly converge less number

frames consid

ors over first story R vectors xcited till rovement

tures has to exact of LDR

dered the

TO 3 F

toThritpabv

The 14th

WoOctober 12-1

3.2. Drift Re

Figure 7 and

the height of of regular struTable 3 showheight of strresponse of important. Usthe obtained parameters, haccurate resubenefits of invectors algori


esponse

Table 2 show

structure incructures there

ws the accuracructure. Howa structure, se of LDR ve

results (exchitherto studiults, even witncluding highithm.

Figure 6 Comp

Figure 7 Co


w the results f

reases, the usis no importacy of the prop

wever, it shoubut in some

ectors for regucept for the ted, for irreguth smaller nuher mode re

parison of story

omparison of drif

hquake Engi

for drift ratio

e of LDR vecant differenceposed method

uld be noted irregularitie

ular frames, dtime of calcuular frames eumber of LDRsponses (via

moment betwee

ft ratio between

ineering

6

( i

ii

h1−Δ−Δ ) o

ctors will leade in using eithd for the “Ovthat higher

s (like the odoes not haveulations and

especially for R vectors usstatic correc

en NLRHA and M

NLRHA and MP

of the irregula

d to better esther LDR vectoverall Drift Inmodes have

one consideree any superior

number of taller frames

ed for the anction concept

MPA using eigen

PA using eigenv

ar structures.

timates of exaors vs. eigenvndex ( /TOP HΔ

e little influeed here) theyrity over the uvectors requs LDR vectornalysis. This t) in the star

nvectors and LD

vectors and LDR

It can be see

act values. In vectors. )” where H ince on disply may becomuse of eigenv

uired). But likrs will produis mainly du

rting vector f

DR vectors

R vectors

en that as

the case

s overall lacement me more vectors in ke other

uce more ue to the for LDR

TO

F R

The 14th

WoOctober 12-1

From the abo

Regular struc

• Best partic

• For mcalcu


Ta

Story

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Tab

Story

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

ove mentioned

ctures:

results are cipation. most of low rulation. As th


able 2 MPA erro


- - - - - - - - - - - -

-13.99 -10-18.98 -17-22.42 -7-45.41 -26

ble 3 MPA erro


- -- -- -- -- -- -- -- -- -- -- -- -

-16.62 -16-18.90 -17-28.84 -13-45.32 -27

d results, the f

obtained usi

rise to mediuhe height of s

hquake Engi

or percentage in

ular 8 StoryDR Eigen - - - - - - - - - - - - - - - - - 8.12 - -14.14- 2.15 - 16.29

0.43 9.23 7.93 -47.657.49 -26.626.92 -38.07

r percentage in d

ular 8 Story-DR Eigen - - - - - - - - - - - - - - - - - 3.80 - 3.66 - 7.25 - 9.00 .01 0.83 .41 -41.57 .15 -32.09 .01 -38.51

following obs

ing first set

um rise regulstructure incr

ineering

7

drift ratio using

-Irregular 12LDR E

- - - - - -1- -1- -1- -1

9.21 8-11.76 12.11 12

16.13 910.48 -3-43.46 -4-18.82 -1-27.43 -3

drift index using

-Irregular 12LDR E

- - - - - 3- 4- 7- 1

3.76 13.50 17.25 89.71 -2.65 -2

-35.98 -1-22.97 -2-27.53 -3

servation can

of LDR vec

lar models ( Hreases, neglec

LDR vectors or

2 Story-Irregularigen LDR - - - - - - - -

8.55 -21.586.00 -15.453.88 -13.80

11.39 -11.558.97 9.15 5.66 16.062.58 12.739.49 9.66 39.74 -37.284.37 2.12 6.99 -6.39

32.77 -17.84

g LDR vectors or

2 Story-IrregularEigen LDR

- - - - - - - -

3.50 3.514.81 4.837.70 7.691.37 11.406.12 16.192.60 12.708.01 8.251.05 -0.27

26.94 -21.5616.42 -6.7623.86 -11.3032.97 -17.64

be made for

ctors with m

3≤BH ) fircting upper m

r eigenvectors

r 16 Story-Irre Eigen L

-9.08 --32.01 -3-21.42 -2-23.45 -2-5.80 -8.92 117.39 114.46 18.61 1.91 2.77 -2.06 -4.30 -9.48 -17.16 -32.40 -

r eigenvectors

r 16 Story-Irr Eigen

2.71 2.56 6.72 8.29

10.90 12.15 12.13 9.76 7.10 4.26 -4.79 -8.06

-12.39 -16.56

-22.58 -32.28

regular and ir

more than 90

rst mode is dmodes may ca

egular LDR 12.66 31.86 20.71 24.19 -6.08 10.08 19.27 15.71 9.12 2.39 6.77 5.84 4.53 5.27 5.28 -2.47

regular LDR 2.95 2.81 6.97 8.61 11.31 12.58 12.50 10.21 7.96 6.72 5.22 5.23 4.50 4.56 3.95 -1.48

rregular struc

0% cumulativ

ominant for rause serious

ctures:

ve mass

response errors in

TO

I

4 T

ubum

R

The 14th

WoOctober 12-1

estimsuper

• Use oslight

• For thlead first f

Irregular stru

• Best massof veeigen

• Use eigencorretheir

• In thfirst s

4. CONCL

The paper prstructures wituse of LDR vbesides accuruse of exact meaningful dselecting the

REFERENC

1. BuildBuild

2. ChintUsingEarth

3. Chopfor bu

4. ChopEd., P

5. GhafAcceVol.

6. WilsoStruc

7. WilsoSupe


mation of the rriority over eiof LDR vectotly underestimhe case of lowto better resufew LDR vec

uctures:

response esti participationectors requesnvectors, so itof LDR vec

n-modes, sincection concepteffects can bee case of usiset of LDR ve

LUSION

roposed an Mth stiffer low

vectors is supracy, are the meigenvectors

differences in number of re

CES

ding Seismicdings, FEMA-tanapakdee, Cg Vertically hquake Enginpra AK, Goeluildings.” EQ

pra AK. (200Prentice Hall,fory-Ashtianyeleration App13, No. 2 andon, E. L., (20ctures, Inc. Beon, E.L., rposition of R


response paraigenvectors (eors (because omated results w rise to med

ults. Since, atctors (dependi

imates are obn. One shouldsted, as this t is important ctors providece the startingt. Consideringe compensateing single groectors with m

MPA proceduwer stories. Fr

erior to use omost importan

for this specusing either

quired Ritz v

c Safety Cou-273 .Federal C. and Chopr

“Regular” neering Reseal RK. (2002)

Q. Eng. and St1). Dynamics, Englewood

y, M. (1989). proach"; Iranid 3. 002), Three Derkeley, Calif

Yuan, M.WRitz Vectors",

hquake Engi

ameters. For except for theof their naturefor higher sto

dium rise regut the end of Ling on the siz

btained using d keep in min

number incto choose ads better estimg vector in Lg the effect oed by the firstound motion,

more than 90%

ure using LDrom the numeof exact eigenvnt factors in scial kind of irset of vector

vectors is a rul

uncil. (1997)Emergency M

ra, A.K. (200and Irregula

arch Center, U“A modal pu

tructural Dyns of StructureCliffs, New J"Seismic Res

ian Journal o

Dimensional fornia, USA. W., Dicken, EQ. Eng. an

ineering

8

this group ofe number of ve) usually leadories compariular models,

LDR vectors ae of the syste

the first set ond that the shcreases shapeequate numbemates of respLDR vector gf higher (rigidt vector of LD, results are m

% cumulative

DR vectors. Uerical results vectors basedselecting an arregularity. Frs. It should ble of thumb a

). NEHRP GManagement 3). “Evaluatioar Generic University of ushover analynamics; 31(3)es: Theory andJersey. sponse of Sixof Science an

Static and Dy

ns, J.M.nd Structural D

f models usinvectors requireds to over estiing to use of eusing more thalgorithm an

em) will conv

of LDR vectoapes of LDRes of first feer of LDR veponse calculgeneration algd) modes to th

DR. more scatteremass particip

Use of the Lit can be see

d modal propeapproach, use or the case obe kept in miand it needs m

Guidelines foAgency: Wason of The MoFrames” ReCalifornia, Bysis procedur: 561-582. d Application

x Correlated End Technology

ynamic Analy

(1982). "DDynamics, vo

ng LDR vectoed in some caimated resultseigenvectors. han 2 or 3 LDeigenvalue perge to the ex

ors with less R vectors are dew vectors wctors. ation in comgorithm is eqhe response o

ed. Best resulpation.

DR vectors, en that for thierties. As the of LDR vect

of regular struind that the ru

more investiga

or the Seismshington, DC.odal Pushoveeport No.PEE

Berkeley. re for estimat

ns to Earthqu

Earthquake Cgy: Earthquak

ysis of Struct

Dynamic Anol. 10, pp 813

ors has no meases). s for lower st

DR vectors wproblem is soxact eigenvect

than 90% cumdependent onwill be the

mparison withquivalent to tof the irregula

lts are obtain

was investigis type of irresimplicity an

tors can supeructures there ule proposed

ation.

mic Rehabilit. er Analysis Pr

EER-2003/03,

ting seismic d

uake Engineer

Components bke Engineerin

tures. Compu nalysis by -821.

eaningful

ories but

would not lved and tors.

mulative n number same as

h use of the static ar frames,

ed using

gated for egularity, nd speed, rsede the were no here for

tation of

rocedure Pacific

demands

ring. 2nd

by Mode ng Issue,

uters and

Direct

use of load depend ent ritz vectors in modal …

Documents